Let $ I $ be a monomial ideal of a polynomial ring $ S $ over a field $ k $. We study the regular elements and zero divisors of $ S/I $, along with the associated primes of $ S/I $ and its quotients by a regular linear form. Using the concept of a vertex cover of a hypergraph, we provide a method that produces a regular linear form on $ S/I $ from an arbitrary regular element of $ S/I $ in the maximal ideal of $ S $. For each associated prime $ P $ of $ S/I $, we exhibit an explicit monomial $ {\bf m} $, expressed in terms of a maximal cover of the exponent vectors of the minimal generators of $ I $, that annihilates $ P $ (that is, $ P{\bf m}\subset I $). Then, we determine which sets of powers of variables can be added to $ I $ to force depth zero in the quotient, leading to a notion of stability that generalizes the classical notion of a stable set in a graph and that is governed entirely by the maximal covers of the exponent vectors of $ I $. Finally, when $ I $ is the edge ideal of a simple graph $ G $ and $ r $ is a regular linear form on $ S/I $, for each minimal vertex cover $ C $ of $ G $, we characterize when the product of the variables outside $ C $ is a nonzero element of $ S/\langle I, r\rangle $ whose annihilator is the associated prime $ \langle C, r\rangle $.
Citation: Jovanny Ibarguen, Daniel S. Moran, Carlos E. Valencia, Rafael H. Villarreal. Regular linear forms of a monomial ideal[J]. AIMS Mathematics, 2026, 11(11): 36275-36292. doi: 10.3934/math.20261405
Let $ I $ be a monomial ideal of a polynomial ring $ S $ over a field $ k $. We study the regular elements and zero divisors of $ S/I $, along with the associated primes of $ S/I $ and its quotients by a regular linear form. Using the concept of a vertex cover of a hypergraph, we provide a method that produces a regular linear form on $ S/I $ from an arbitrary regular element of $ S/I $ in the maximal ideal of $ S $. For each associated prime $ P $ of $ S/I $, we exhibit an explicit monomial $ {\bf m} $, expressed in terms of a maximal cover of the exponent vectors of the minimal generators of $ I $, that annihilates $ P $ (that is, $ P{\bf m}\subset I $). Then, we determine which sets of powers of variables can be added to $ I $ to force depth zero in the quotient, leading to a notion of stability that generalizes the classical notion of a stable set in a graph and that is governed entirely by the maximal covers of the exponent vectors of $ I $. Finally, when $ I $ is the edge ideal of a simple graph $ G $ and $ r $ is a regular linear form on $ S/I $, for each minimal vertex cover $ C $ of $ G $, we characterize when the product of the variables outside $ C $ is a nonzero element of $ S/\langle I, r\rangle $ whose annihilator is the associated prime $ \langle C, r\rangle $.
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Figures(1)
Jovanny Ibarguen, Daniel S. Moran, Carlos E. Valencia, Rafael H. Villarreal. Regular linear forms of a monomial ideal[J]. AIMS Mathematics, 2026, 11(11): 36275-36292. doi: 10.3934/math.20261405
Graph $ G $ such that $ S/I_G $ has depth three, and which admits two regular linear forms $ r_1 $ and $ r_2 $ with disjoint support such that $ r_2 $ is not regular in $ S/\langle I, r_1\rangle $
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